Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{p^2 - 10p + 16}{p^2 - 8p}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{p^2 - 10p + 16}{p^2 - 8p} = \dfrac{(p - 2)(p - 8)}{(p)(p - 8)} $ Notice that the term $(p - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p - 8)$ gives: $q = \dfrac{p - 2}{p}$ Since we divided by $(p - 8)$, $p \neq 8$. $q = \dfrac{p - 2}{p}; \space p \neq 8$